Let G denote a closed, connected, noncompact subgroup of GL(n,R) that is self-adjoint (invariant under the transpose operation). Let d_{R} denote the right invariant Riemannian distance function defined by the Euclidean inner product on M(n,R). We study a kind of metric entropy for group actions in which the real numbers R are replaced by a Lie group G and Diff(R^{n}) is replaced by GL(n,R).
More specifically, for a fixed vector v in R^{n} we obtained algebraically defined upper and lower bounds L^{-}(v) and L^{+}(v) for the asymptotic growth rate of the function g → log |g(v)|/d_{R}(g,G_{v}) as d_{R}(g, G_{v}) → \infty, where G_{v} denotes the subgroup of G that fixes v. If G_{v} is compact, then we may replace d_{R}(g, G_{v}) by d_{R}(g,Id) in the result above.
We compute L^{-}(v) and L^{+}(v) for the irreducible, real, linear representations of G = SL(2,R). If the dimension of the G-module V is odd, then L^{_}(v) = L^{+}(v) for all v in a nonempty open subset O of V. In particular, the function g → log |g(v)|/d_{R}(g,G_{v}) has a limit for vectors v in O as d_{R}(g, G_{v}) → \infty. These examples show that the functions L^{-}(v) and L^{+}(v) are SO(2,R)-invariant but not SL(2,R) invariant.